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Orifice Plate Critical Pressure Ratio

Orifice Plate Critical Pressure Ratio

Can anyone advise how to determine the Critical Pressure Ratio at which a Square Edge Orifice Plate with compressible fliud reaches critical (sonic) flow and the flows at pressure ratios beyond the critical ratio.

I need to correctly predict flowrates through pressurising and blowdown restriction orifices from high pressures where dP/P1 ratios are as high as 0.98 and the readily available texts do not provide the solution.

Almost all of the orifice plate calculations identified are for accurate measurement flows below the critical range and  limit dP/P1 to say 0.35 or 0.6 to ensure the flow is subsonic.

Crane TP410 provides the formula for critical flow but first the Net Expansion Factor Y chart on A-21 is only provided up to dP/P1 = 0.6 and second the Critical Pressure Ratio chart on the same page is only provided for Nozzles and Venturi Meters and not for Square Edge Orifices.

I understand that the flow will continue to rise as the downstream pressure falls further because the vena contracta grows in diamter.

Dennis Kirk Engineering

RE: Orifice Plate Critical Pressure Ratio

Hi dennis let me have ur e-mail & I'll send u a good spreed sheet (for oriffice), I hope it will help:

RE: Orifice Plate Critical Pressure Ratio

I've calculated the equivalent K for an orifice (the approximate formula is in Crane) and then used the charts in Crane to estimate the maximum pressure ratios.  The problem is that for large K values, which orifices usually work out to, you have to guess (sorry, estimate ) at a dP/P1 max value.

But, there was a recent article in Chemical Engineering on compressible flow I'd look up.  It was in the last few months and included a spreadsheet that extended the curves in Crane significantly.  I've checked it against my own spreadsheets and got fairly similar results.

RE: Orifice Plate Critical Pressure Ratio


You may want to look at "Flow Measurement Engineering Handbook", R.W.Miller, McGraw-Hill.  It has a chapter on critical flow and covers discussion on ISO critical venturi nozzles, ASME long-radius nozzles, and square-edged orifices.

To quote directly from that reference, "Cunningham (1951) first drew attention to the fact that choked flow will not occur across a standard, thin, square-edged orifice." As an additional quote, "However, choked flow is achieved by increasing the plate thickness."  Keep this in mind when you specifiy your orifice plate.

Without going into a lot of detail, the critical pressure ratio takes the usual form of

p2/p1 = ( 2 / (k+1) )^( k / (k-1) )

There is an additional term (not shown) accounting for correcting static pressure to total pressure.  I think in a lot of instances, that term can be taken as 1.  I suggest you check the above reference for full details, it has some worked example problems for critical flow orifices as well.


Seems I've seen mention of the article in Chem Engineering regarding compressible flow before but I haven't had much luck locating it.  Any chance you could provide more info about locating the article?  Thanks, in advance.  Ellis

RE: Orifice Plate Critical Pressure Ratio

Thanks for the responses I will work on this information and look forward to more suggestions. When I have it resolved I will post my findings in this thread.

My email is

I have examined several calculations (including my own) for flow through restriction orifices and am convinced that they are either very rough or are incorrect. There seems to be a hole in the text book thoery on this topic?

Dennis Kirk Engineering

RE: Orifice Plate Critical Pressure Ratio

Critical pressure drop is dependant on beta ratio and specific heat ratio.

The actual DP will not change flow rate once past the critical ratio. I'm guessing this means that you should use the expansion factor determined at the critical pressure ratio for calculating flow?

RE: Orifice Plate Critical Pressure Ratio

The lack of true critical flow behavior with any sharp-edge restriction is well known, but almost impossible to characterize exactly. The entrance losses are the main problem.

Millers book is the best resource and one that is available.

Several manuf. do make chokes (nozzles) for this purpose to over come the limitations of restriction plates.


RE: Orifice Plate Critical Pressure Ratio

hacksaw - Thank you. I have been reading about this today and three significant texts indicated as you have, that thin sharp edged orifices do not exhibit limiting (critical) flow. The throughput continues to rise as the downstream pressure is reduced (and the vena contracta area increases).

I still find it hard to believe because eventually the throat must approach the size of the orifice hole and the flowrate stop increasing?

OK so how do you determine the flow through an orifice plate at high dP/P1 ratios? Is there a way to determine the vena contracta area or Cd (Cd is supposed to be the ratio of Av/Ao) because surely the velocity in the throat must be sonic when the pressure ratio P2/P1 exceeds the magic number of 0.528?

There must be an answer out there somewhere as flow at high dP seems to be such a common thing (restriction orifices, PSVs, bursting disks) that it cannot be unsolved!

Dennis Kirk Engineering

RE: Orifice Plate Critical Pressure Ratio

You only get sonic flow conditions in ideal converging nozzles e.g. "critical flow" nozzles used a flow calibration standards. Never the less you will find thick plates (T/D>>1) used in flow limiting services. Thisck plates not equal to a flow nozzle in terms of holding the flow constant, but they are rugged and work well enough for plant applications.

Restriction orifice plates are an expeditious measure to limit the flow not hold it constant. High drop applications really have not been studied simply because of the problems that arise with plate distortion (bending). To answer your specific question: isentropic expansion across a thin plate restriction is not possible and empirical testing would be required.  

Use of "low DB" plates i.e. multi-hole restrictions fall in the thick plate category (T/d>>1) and are well characterized.

Rupture disks and relief valves are all characterized in actual flow testing.

You will notice that relief and safety valves have a contoured nozzles to reduce inlet losses and to improve the discharge velocity during relief.

Another issue with R.O. sizing is that you have to size them on the basis of un-recovered loss not developed d/p adjacent to the plate.

RE: Orifice Plate Critical Pressure Ratio

EGT01, sorry for the delay in answering your question.  

The article on compressible flow was in the October 2002 edition of Chemical engineering magazine.  I have been looking for without any success for my copy (though I know I had it) and in addition, I've been on the road since you posted your question hence the delay.

I do have a copy of the spreadsheet that the author made freely available for downloading.  I can forward you a copy of that if you like, if so, drop me a line at  You can also download it at  Sorry, don't have a more extensive link than that.

RE: Orifice Plate Critical Pressure Ratio

Could you try that link again to the article and spreadsheet you mentioned. Otherwise is it OK to get a copy from you.

RE: Orifice Plate Critical Pressure Ratio


Thanks much for the reply.  I was able to download from the links you provided.

RE: Orifice Plate Critical Pressure Ratio

TD2K and Denniskb and Others:

When one tries to size a sween, natural gas pressurized piping/vessel system that is to be blowdown to atmosphere, one often sees an inlet on/off valve, a variable-position pressure reduction regulator (or fixed orifice), a long length of one, two or three different diameters, a second fixed orifice immediately upstream of, or a part of, a silencer, and of course a silencer.

One could start with a nominal pressure at the silencer and work backwards for a given flow, calculate the backpressure that should exist downstream of the regulator (or first orifice).  Then, one could state that this is the pressure drop required of the regulator valve (or first orifice).  

However, as the vena contracta expands (and filling up that pipe), isn't there some or a lot of pressure recovery?  Such that, the pressure that exists several feet downstream of the vena contracta is HIGHER than the pressure that exists by calculation in the middle of the vena contracta?

Any ideas on how to approach a system calculation like the above?

RE: Orifice Plate Critical Pressure Ratio

pressure recovery very real, but not where you have critical flow and irreversible processes.

it is a straight forward calculation, but not necessarily one that you would perform by hand.

RE: Orifice Plate Critical Pressure Ratio

Hookem and other contributors,

The solution method for the blowdown system with many components is simple so long as the flow does not go critical at any point. Simply guess a flow and work forward or backward calculating the pressures at each step. When you get to the end if the calculated pressure does not match that required then simply adjust the flow and run it again until the calculated pressure matches the actual pressure.

It fact the flow can go critical at many points and often at more than one at a time.

Pressure recovery is simply the conversion of kinetic energy to pressure based on the change of velocity from the orifice to the mean downstream pipe velocity and is easy to calculate.

If sonic flow occurs at only one location within the path then you must calculate forward to that device from the inlet AND back to that device from the exit.

As is pointed out several times above it is apparent that there will be a limiting flow for an orifice and it should be predictable. Despite the efforts we were not able to find a way to determine the vena contracta area other than some suggested typical values.

The closest we came to finding a solution was from the optimal-systems web site for the GasDp software. Unfortunately we were unable to contact the author or track down his software. In his software sample print out he has treated the orifice plate as a very short (1mm) length of pipe, with entry losses due to the size change, discharging into a larger area. The method seems to follow the same approach as Crane page A-22 which provides limiting vales for Y and dP/P1. This suits me as this is exactly where I expected to end up and is the solution I intend to implement in my own calculation.

It would have been nice to find this solution elsewhere as well so it had more support but with the lack of another solution I will adopt it for now.

Dennis Kirk Engineering

RE: Orifice Plate Critical Pressure Ratio

I found a useful reference from the RW Miller handbook. Try A.J. Ward-Smith "Critical Flowmetering: The Characteristics of cylindrical nozzles with Sharp Upstream edges" Int J Heat Fluid Fl vol 1 No 3 pp 123-132 1979

In my case I needed to predict the flow vs pressure characteristic of a condenser sparger pipe, inlet P of about 100 psia, outlet P about 2 psia, 0.75" dia holes in a 0.25" thk rolled plate ( t/d= 0.33)

Per Ward-Smith, the choked compressible flow Cd is a function of the t/d ratio. As follows:
sharp edge, t/d= 0, Cd = 1.0
thin plate (0<t/d<1)Cd varies smoothly from 1 to 0.81 as function of t/d.
thick plate ( 1<t/d<7) Cd = 0.81 constant
very thick plate (t/d > 7) Cd less tahn 0.81 per Fanno friction

Tremendous difference compared to standard non choked sharp edge Cd of 0.64

RE: Orifice Plate Critical Pressure Ratio

The biggest source of inaccuracy results from errors in estimating:
(1) k (i.e., Cp/Cv ratio), and
(2) z, the gas compressibility factor, used in the gas density calculation.

For a real gas, both need to be found using a respectable equation of state, such as Lee-Kesler or Peng-Robinson.

NOTE: The common procedure often used to find Cp using ideal gas properties, then finding Cv using Cv = Cp - R, a result valid only for ideal gases, can be quite inaccurate unless you are below about 3 atmospheres pressure and well above the mixture critical temperature.

Miller's "Flow Measurement Engineering Handbook" (3rd edition) provides the best summary of the orifice equations.

If you wish, I can help with these properties - email me at with your mixture composition and system T&P.

RE: Orifice Plate Critical Pressure Ratio

EGT01 notes a reference from Cunningham, 1951.  The article name is "Orifice Meters With Supercritical Compressible Flow".  It is on pages 625 through 638 of the July 1951 issue of "Transactions of the ASME".  This is the best general reference on this subject I am aware of.  I can scan for you if you cannot find in reference libraries.

RE: Orifice Plate Critical Pressure Ratio

Statoil (Norway) uses API 520 (Part 1) for sizing of orifices, utilizing the same equation as for PSV's (which really just is an orifice once lifted - correct me if I'm wrong). See API 520, Chap. 3.6, which provides equations for critical pressure ratio, and sonic and subsonic flow.

However API 520 specifies a discharge coefficient Kd of .975, which seems to be too high. We recommend using a factor of .81 for blowdown.

What confuses me however, is that there seems to be some confusion around the discharge factor.  Which, as far as I can see, is not the same as the one used in e.g. Crane.

If anyone can help me with a good source on the API discharge factor, I'll be grateful.

I can also recommend the neat little shareware program Orifice from (free to try for 1 month), which does both critical and non-critical flow.


RE: Orifice Plate Critical Pressure Ratio

The paper referenced above, by Ward-Smith, shows that a Cd = 1.0 can be ustilized in the theroretical case of t/d=0, so a knife edge might  use Cd= 0.97. For thicker plates and after erosion, a Cd=0.81 wold be recommended.

RE: Orifice Plate Critical Pressure Ratio


I have been reading up on this topic now, and I must say your postings have helped me a lot.

Quote from an earlier post:

"thin plate (0<t/d<1)Cd varies smoothly from 1 to 0.81 as function of t/d."

By smoothly, do you mean linear, or is there a function?

If so, could you please post it?


RE: Orifice Plate Critical Pressure Ratio

The case of t/d is practically impossible to meet , even a razor edge brand new the Cd= 0.97.

The curve suggests a semi log relationship  of Cd(linear) vs log,10 (t/d)where Cd=0.97 at t/d = 0.01 and Cd= 0.81 at t/d=1.0

RE: Orifice Plate Critical Pressure Ratio

Thanks davefitz

Cd = 0,81 - 0,0347 Ln(t/d)
for square edged orifices with t/d < 1

ok with you?


RE: Orifice Plate Critical Pressure Ratio

OK, but you should get a copy of the paper and review fig 7 and the tables of test data to get an idea of the accuracy.

I had been using a Cd=0.84 for a t/d = 0.5;  a better plot on semi log paper would use the 2 points t/d=1, Cd = 0.81 and t/d=0.5 Cd=0.84.  There iia lot of data that supports the Cd=0;84 at t/d=0.5

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